Acta mathematica scientia, Series B >
LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS
Received date: 2009-11-06
Online published: 2009-11-20
Supported by
Temple supported in part by NSF Applied Mathematics Grant Number DMS-040-6096. Young supported in part by NSF Applied Mathematics Grant Number DMS-010-4485.
In this paper we show how the simplest wave structure that balances compression and rarefaction in the nonlinear compressible Euler
equations can be represented in a solution of the linearized compressible Euler equations. Such waves are exact solutions of the
equations obtained by linearizing the compressible Euler equations about the periodic extension of two constant states separated by
entropy jumps. Conditions on the states and the periods are derived which allow for the existence of solutions in the Fourier 1-mode.
In[3, 4, 5] it is shown that these are the simplest linearized waves such that, for almost every period, they are isolated in the kernel of the linearized operator that imposes periodicity, and such that they perturb to nearby nonlinear solutions of the compressible Euler equations that balance compression and rarefaction along characteristics in the formal sense described in[3]. Their fundamental nature thus makes them of interest in their own right.
Key words： compressible Euler; periodic solutions; conservation laws
Blake Temple , Robin Young . LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS[J]. Acta mathematica scientia, Series B, 2009 , 29(6) : 1749 -1766 . DOI: 10.1016/S0252-9602(10)60015-X
[1] Craig W, Wayne G. Newton's method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math, 1993, 46: 1409--1498
[2] Smoller J. Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag,1982
[3] Temple Blake, Young Robin. A paradigm for time-periodic sound wave propagation in the compressible Euler equations. Methods Appl Analy, 2009, to appear.
http://www.math.ntnu.no/conservation/2008/033.
[4] Temple Blake, Young Robin. Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors. SIAM J Math Anal, 2009, to appear.
http://www.math.ntnu.no/conservation/ 2008/034.html.
[5] Blake Temple, Robin Young. A Liapunov-{S}chmidt reduction for time-periodic solutions of the compressible Eler equations. 2009.
http://www.math.ntnu.no/conservation/2008/035.html
[6] Young Robin. Global wave interactions in isentropic gas dynamics. Submitted, 2008.
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